The Weak Galerkin Finite Element Method For Solving Two Types Of PDEs

Weak Galerkin finite element method(WGFEM)is a new and efficient numerical method for solving partial differential equations.It is the inheritance and development of standard finite element method,which is first proposed by Wang and Ye for solving the second-order elliptic problems in 2011.Compared with the classic finite element method,the WGFEM introduce the weak function and the weak differential(such as weak derivative,weak gradient,weak divergence,weak curl etc.).These concepts are involved in the variational form for solving the partial differential equation.After it was proposed,the WGFEM received wide attention.Over the past few years of development its content is more abundant.These developments can be summarized as follows:the introductions of mix form and stabilizer enable the WGFEM more practicable;the solver is more robust with the hybrid and adaptive technology;the multiscale idea is introduced into the WGFEM,which makes it possible to handle multi-scale physical and engineering problems.With the improvement of research content,this method can deal with more and more problem.Up till now,many researchers have adopted the WGFEM for solving a number of classic problems,such as the second-order elliptic interface problem,the Biharmonic equation problem,the Helmholtz equation problem,the Stokes equation problem,and the Maxwell equation problem etc.In this paper,we study the one dimensional grating diffraction problem and Navier-Stokes equations.In order to solve the difficulties of two kinds of problems,some processing techniques are presented and the corresponding weak Galerkin finite element scheme are designed.There are two difficulties in grating diffraction problem.The first one is that the computational domain is unbounded region.We need to truncate the unbounded domain and give the corresponding boundary conditions which affects the results directly.Another issue of diffraction grating problem is that the gradient of the solution changes rapidly near the interface.The traditional finite element method can not capture the shock or jump near the interface well.It is necessary to search for more stable and more accurate numerical methods.For the first difficulty,we use periodic boundary condition for x1 direction,and transport boundary condition for x2 direction for the one-dimensional periodic grating problem.The the computational region is truncated into a rectangular region[0,A]x[-b,b].Here A denotes the period for x1 direction and b denotes a positive constant.Then,we solve the Helmholtz equation with periodic and transport boundary conditions as following where the transport boundary condition in x2 direction is given by Here,the operators ?? and Tj,j = 1,2 are defined in(3.3)and(3.8),respectively.For the second difficulty,we use the WGFEM with a stabilizer to solve the above equation.Due to weak differential and the stabilizer,the numerical solution is continuous weakly.It is very suitable to simulate electromagnetic waves in the vicinity of the grating interface.Based on the above discussion,we verify the reliability of the algorithm by theory and numerical experiment,respectively.In theory,the weak Galerkin finite element scheme with transmission boundary conditions and periodic boundary conditions is analyzed in detail.The existence,uniqueness and convergence of the solution are given,and the order of convergence is optimal.The error estimates are given by the following theorem:Theorem Let u?Hm+1(?)and uh?Vh be the exact solution of(3.5)-(3.7)and the weak Galerkin solution of(3.15)respectively.Deenote eh=Qhu-uh={ei,eb),and assume the dual problem for the grating problem(3.5)admits a solution ??H1+8 with s?(0,1),,then there exists a costant h*>0 such that hold for any h?(0,h*).At present,the convergence analysis of WGFEM is mainly for Dirichlet boundary conditions.However,the theoretical results with the transmission boundary condition are very few.The theoretical work in this paper enriches the contents of WGFEM and it has some reference value for other similar work.In numerical,we simulate three grating diffraction problems.The accuracy of the algorithm is verified from convergence and grating efficiency,respectively.About conver-gence,the error convergence rates of ?·?Wh-norm and L2-norm are 1 order and 2 order,respectively.It coincides with theoretical results.About grating efficiency,the sum of the reflection efficiency and the transmission efficiency is near 1,which coincides with the fact.Moreover,we also give the diffraction efficiency of main diffraction order and high oscillatory function image of electric field intensity.From three numerical simulations,we find that WGFEM has following advantages in solving the diffraction problem:firstly,the algorithm is stable,efficient and highly accurate for both simple and complex gratings;secondly,the solution of WFFEM is continuous weakly.It is very suitable to capture the oscillation of electromagnetic wave near the grating interface.For the computational fluid dynamics problem,consider an bounded domain ?(?)R2 with a Lipschitz boundary ??(?)?.The velocity u and the pressure p of the Navier-Stokes equations for incompressible viscous flows are governed by ut-v?u+(u·?)u-?p=f,in ?×(0,T],? · u = 0,in ?×(0,T],u = g,on ?×(0,T],u(·,0)= u0,in ?,where v,f and u0 denote viscosity parameter,source term and initial data,respectively.g is the boundary condition satisfying ??g·nd?=0,n denotes the unit vector of outward normal at ?,and T>0 is the terminal time.An important concept for this problem is the Reynolds number(Re=UL/v)which is the similarity criterion of viscosity in fluid mechanics,where U denotes velocity magnitude and L denotes the size of the region.There are three difficulties for solving the Navier-Stokes equations.Firstly,When Reynolds number is high,classic finite element method is not work for the pressure;the second difficult is how to choose the approximation spaces for the velocity and pressure;the another difficult is the nonlinear term in the Navier-Stokes equations which makes theoretical analysis and solving numerically more difficult.For the first difficult,Weak Galerkin finite element method is an effective treatment method.This method introduces a stabilizer to control the jump between the inside and the boundary of each element.With the help of stabilizer,Weak Galerkin finite element method scheme is more stable so that the pressure can be simulated well.The parameter for stable term can be chosen easily according to relative high Reynolds numbers,whereas the stabilized FEMs always needs to adjust the parameters for numerical flux terms or penalty terms carefully to guarantee the stability of numerical method.The second difficult is the choice of velocity and pressure approximation space in solving Navier-Stokes equations numerically.If the finite element spaces for velocity and pressure doesn't match,the numerical results suffer from instabilities or highly oscilla-tory pressure field.There are two categories strategies to prevent this phenomena from happening:the mixed finite element methods and the stabilized finite element method.In the former strategy,the degree of piecewise polynomial in finite element spaces for pressure field is always chosen as one order lower than the degree of piecewise polynomial for velocity field,in order to satisfy inf-sup condition.In the latter strategy,the Galerkin least-squares(GLS)stabilization is adopted,where the stabilizing terms are given by min-imizing the squared residual of the momentum equation over each element.Comparing with mixed finite element method,the Weak Galerkin finite element method has more choices of combination for the order of the velocity and pressure.For example,one can either choose local polynomial Pk-Pk-1 for velocity and pressure respectively,or a equal order,even more complicated combinations Pk-Pk We shall show all these choices satisfy the inf-sup condition under a mild assumption later and check their performance numeri-cally.Comparing with the stabilized finite element method,it is easier to select parameters for weak Galerkin finite element method.For the last difficult,the nonlinear term brings great difficult to both theoretical analysis and numerical calculation.In the aspect of theoretical analysis,we need more technique to derive the well-posedness of the numerical scheme;in the aspect of numerical calculation,the nonlinear term make program more complex and we needs more time to get the result.This paper uses Weak Galerkin finite element method solving Navier-Stokes equations,and the existence,uniqueness and convergence of the solution of semi-discrete Weak Galerkin finite element scheme are given.we adopt the Newton iterative method with tolerance 10-7 to deal with nonlinear term.Because of the stability of Weak Galerkin finite element method,Newton iterative method needs only 2-4 iterations to achieve convergence.For the velocity and pressure,we have error estimate as following:Theorem Let {u,p} and uh be the solution of NSEs(4.1)-(4.4)and(4.28),respec-tively.Let the assumption(4.32)hold,then we have the following estimatewhere C is a constant independent of h,Qi and Qh are defined by(2.26)and(2.26),respectively.Theorem Let {u,p} and ph be the solution of NSEs(4.1)-(4.4)and(4.29),respec-tively.Let the assumption(4.32)hold,then we have the follouing estimatewhere C is constant independent of h,Ph is defined by 4.18).To sum up,we apply the WGFEM for solving diffraction grating problem and Navier-Stokes problem in this thesis,where both theoretical results and numerical simulations verify the efficiency of our algorithm.